The Euler-Lagrange Cohomology Groups on Symplectic Manifolds

2003/04/20 by Han-Ying Guo, Guo, Han-Ying, Jianzhong Pan +5
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Classical Physics (physics.class-ph) #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #physics.class-ph

paper · pdf · doi:10.48550/arxiv.physics/0304074

20 pages, no figures

arxiv created 2003/04/20 · openalex publication_date 2003/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The definition and properties of the Euler-Lagrange cohomology groups H^2k-1, 1 \leqslant k \leqslant n, on a symplectic manifold (\cal M²ⁿ,ω) are given and studied. For k = 1 and k = n, they are isomorphic to the corresponding de Rham cohomology groups H_dR¹(\cal M²ⁿ) and H_dR^2n-1(\cal M²ⁿ), respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomology groups on (\cal M²ⁿ,ω), in general. The general volume-preserving equations on (\cal M²ⁿ,ω) are also presented from cohomological point of view. In the special cases, these equations become the ordinary canonical equations in the Hamilton mechanics. Therefore, the Hamilton mechanics has been generalized via the cohomology.

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